A study on the crack detection in beams using linear and nonlinear normal modes

Abstract

Linear and nonlinear normal mode motions may provide promising information about the condition of mechanical structures under small and large amplitude vibrations, respectively. In this view, this study investigates the nonlinear dynamics of cracked beams through use of the nonlinear mode motion and extends the crack identification methods that utilize the linear characteristics to nonlinear vibrating structures. At first, the nonlinear normal modes of the intact and cracked beams are calculated by a continuation algorithm. A finite element model of a geometrically nonlinear prismatic beam was created based on crack stress intensity. Subsequently, a method based on normal mode motion and minimization of strain energy, which is valid for linear and nonlinear vibrating beams, was developed as an optimization problem. To this end, hybrid optimization was also used due to its capability in finding global minimum along with its computational efficiency. It was shown that the proposed crack detection technique is applicable to beams vibrating in linear and/or nonlinear regimes and well capable of detecting both crack location and severity.

Keywords

crack detection cracked beams hybrid optimization nonlinear normal modes nonlinear vibration strain energy minimization

Introduction

The use of dynamic characteristics in assessing the current state of mechanical systems has taken a lot of attention. Linear properties such as frequencies and modal shape have been widely used in damage detection and crack identification studies. However, their extensions to nonlinear vibrating structures remain uncommon. Recently, the concept of nonlinear normal modes (NNMs), which can be used to comprehend the nonlinear dynamics of civil and mechanical structures, was introduced. These vibration properties involve valuable information about the current state of the mechanical systems, which may be used for identifying the existing defects in nonlinear vibrating structures as an extension to methods of linear case.

Crack detection/identification problem has been a challenging one and taken considerable attention in the last two decades. These methods mostly rely on the vibration characteristics measured in forced and/or free vibration experiments (Fan and Qiao, 2011; Grip et al., 2017; Yan et al., 2007). In most studies, natural frequencies and modal node displacements have been used as damage/crack indicators. The discrepancies in the modal parameters of intact and damaged structures have been used to detect damage location and magnitude (Esfandiari et al., 2013; Goldfeld and Elias, 2013; Huang et al., 2018; Stache et al., 2016). Some research realized that the modal curvatures are quite sensitive to damage (Ciambella and Vestroni, 2015; Janeliukstis et al., 2017; Yang et al., 2017). As a special case of modal curvature methods, modal strain energy methods have come forth in damage detection literature (Baneen and Kausar, 2018). It was shown that the modal strain energy methods are well capable of finding damage location (Cha and Buyukozturk, 2015; Xu and Wang, 2017; Yan et al., 2012). Commonly, damage/crack identification problems have been considered as the inverse problem of mechanics. That is to say, global and local optimization methods have been utilized in order to find damage/crack parameters by minimizing an objective function (Miguel et al., 2012; Meruane and Heylen, 2011; Perera and Ruiz, 2008). Application of hybrid versions of these optimization methods in damage detection problem was studied in Moezi et al. (2018) and Chen and Yu (2018).

Although the damage detection studies based on linear vibrational characteristic are vast, their extensions to nonlinear vibrating structures are rare. Tsyfansky and Berenevich (1998) presented a numerical study that uses signal from the geometrical nonlinearity of a bar to detect fatigue cracks by vibration monitoring. Rezaee and Shaterian-Alghalandis (2018) proposed a new crack detection methodology by analyzing geometrically nonlinear vibrations of a cracked beam under harmonic excitations. They introduced a crack index parameter, which is sensitive to the existence of cracks. Andreaus and Baragatti (2011) used some quantitative and qualitative parameters obtained from the Fourier spectrum of a harmonically excited beam to relate nonlinear resonances to crack presence, location, and depth. The application of the method to a laboratory structure with an edge crack can be found in Andreaus and Baragatti (2012). A crack detection procedure that is based on the nonlinear response of a beam with breathing edge crack was proposed by Chatterjee (2011).

Recently, the concept of NNMs, which are relevant dynamic features of nonlinear vibrating systems similar to linear normal modes (LNMs) of linear vibrating systems, has taken great attention. Several aspects of NNMs and their development together with algorithm to numerically calculate NNMs were introduced in Kerschen et al. (2009) and Peeters et al. (2009). Computationally inexpensive methods and reduced order models that can be used to compute NNMs can be found in Kuether et al. (2017), Kuether and Allen (2014), and Sombroek et al. (2018). Although numerical computation of NNMs is difficult, identification of NNMs experimentally might be more challenging. A practical experimental methodology using the force appropriation technique was presented by Peeters et al. (2011a, 2011b) to extract the experimental NNMs. An approach that relates the given forced response to the NNM of mechanical systems was given by Liao (2016). In Denis et al. (2018), an experimental strategy to identify the backbone curve of nonlinear modes of mechanical structures can be found. Chang and Poon (2010) showed that the intrinsic mode functions obtained from the empirical mode decomposition method that is used in nonlinear identification are numerically close to NNMs.

In this article, a crack detection methodology has been introduced, which can be used to detect both crack location and severity in linear and geometrically nonlinear vibrating beams with sufficient accuracy. LNMs and NNMs of an undamped vibrating beam have been utilized in this study as the damage-sensitive quantities. Two objective functions have been introduced of which the first one is based on the periodic solutions of conservative mechanical systems and the second one is based on the minimization of strain energy. It was shown that the second objective function is sensitive to crack location, while it is also computationally inexpensive both in linear and nonlinear cases. Besides, the first objective function is sensitive to both crack location and severity, yet, it is computationally expensive in comparison to the first one due to the requirement of time integration. Hence, the second objective function was used as the crack locator and the first one was used to determine the crack severity. The results showed that LNMs and NNMs are promising measurable vibration characteristics that can be used as crack indicators for linear and nonlinear vibrating mechanical structures, respectively.

Numerical calculation of NNMs

In this section, computation of NNMs using pseudo arc length continuation introduced in Peeters et al. (2009) is briefly explained. NNMs can be computed based on the periodic solutions of a conservative system. In this context, equation (1) can be written considering that the difference between the initial conditions $z_{p 0}$ and the system response at the time of minimal period T is zero

H (z_{p 0}, T) = z_{p} (z_{p 0}, T) - z_{p 0} = 20

(1)

where $H (z_{p 0}, T)$ is called the shooting function, and $z_{p} (z_{p 0}, T)$ is the periodic solution of the system. This equation requires time integration over the period T, which can be carried out using conventional time integration techniques such as Newmark scheme. At this point, the initial conditions $z_{p 0}$ and the period T have to be found that satisfy equation (1). Hence, an iterative method that uses the Newton–Raphson algorithm, which starts from a known solution, can be used. Using Taylor series and neglecting higher order terms equation (2) can be obtained

H (z_{p 0}^{0}, T^{0}) + {\frac{\partial H}{\partial z_{p 0}} |}_{(z_{p 0}^{0}, T^{0})} Δ z_{p 0}^{0} + {\frac{\partial H}{\partial T} |}_{(z_{p 0}^{0}, T^{0})} Δ T^{0} = 20

(2)

By solving increments from equation (3), initial conditions and period in the next iteration can be computed by

z_{p 0}^{k + 1} = z_{p 0}^{k + 1} + Δ z_{p 0}^{k} and T^{k + 1} = T^{k} + Δ T^{k}

(3)

where k denotes the iteration index. Convergence is achieved if the relative error is equal to a prescribed precision

\frac{‖ z_{p} (z_{p 0}, T) - z_{p 0} ‖}{‖ z_{p 0} ‖} = ε

(4)

Equation (2) involves 2n equations with 2n + 1 unknowns, where n is the degree of freedom (DOF) of the system. Hence, using phase conditions, additional equations $h (z_{p 0})$ can be introduced. In this study, the initial velocities are taken zero as additional equations

F (z_{p 0}, T) = (\begin{matrix} H (z_{p 0}, T) \\ h (z_{p 0}) \end{matrix}) = 0

(5)

In fact, there are different resonance modes at each energy level due to the frequency–energy dependence (e.g. amplitude dependence) of geometrically nonlinear systems. Hence, equation (5) follows a curve called branch. In other words, equation (5) has multiple solutions for the modal parameters of an NNM for each energy level. Therefore, continuation algorithm has to be used to determine the NNMs at each energy level starting from a known solution. A good starting point (e.g. $i = 0$ ) for the continuation is the underlying LNMs obtained from the conventional eigenvalue analysis of the linear system (e.g. assuming small displacements). Then, a predictor–corrector procedure is applied. In the predictor step, an initial guess for (i + 1)th point ${\bar{w}}_{i + 1} = [z_{p 0, i + 1}, T_{i + 1}]$ in the solution curve is made using the tangent vector $v_{i} = [v_{z_{p 0}, i}, v_{T, i}]$ obtained from the solution of the homogeneous system of linear equations

\nabla F (z_{p 0, i}, T_{i}) v_{i} = 0

(6)

{\bar{w}}_{k + 1} = w_{i} + h v_{i}

(7)

where h is the step size parameter. The norm of predictor vector $v_{i}$ has to be unity. In the corrector step of the (i + 1)th point in the solution curve, the overdetermined linear system of equation given by equation (8) can be solved for $w_{i + 1}^{k + 1}$ Moore–Penrose matrix inverse

[\begin{matrix} {\frac{\partial F}{\partial w} |}_{w_{i + 1}^{k}} \\ {v_{i}}^{T} \end{matrix}] {Δ w}_{i + 1}^{T} = [\begin{matrix} - F (z_{p 0, i + 1}^{k}, T_{i + 1}^{k}) \\ 0 \end{matrix}]

(8)

The presented procedure is straightforward and easy to apply. However, it can be computationally expensive for large-scale problems (e.g. problems containing structural systems with too many DOFs), since calculation of the derivatives with respect to $z_{p 0}$ has to be performed using finite difference if the nonlinearities are not localized. In order to reduce computational burden time integrations can be performed over half period $T / 2$ . In addition, if zero phase conditions are adopted, the velocities can be eliminated from the unknowns, which lead to n + 1 unknowns instead of 2n + 1 unknowns. The use of high performance computers and parallel programming techniques can also help to reduce the computational difficulties.

Cracked beam finite element model

A geometrically nonlinear finite element (FE) model of the cracked beam was developed by extending the work of Viola et al. (2002) in which a linear cracked beam element model was proposed. The presented formulation is restricted to the flexural case and Euler–Bernoulli beam. The beam is modeled with two segments connected rigidly in transversal and axial directions and an elastic hinge having a constant K given in equation (9), where b, h, and $β$ are the beam thickness, height, and thickness to height ratio. The function $f (ξ)$ depends on the crack depth (Viola et al., 2002)

K = \frac{E h^{3} β}{6 f (ξ)}

(9)

The cubic displacement shape $u^{k}$ of the kth segment of the cracked beam is given as a function of beam coordinate $x_{k}$ in equation (10)

u^{k} (x_{k}) = a_{k} + b_{k} x_{k} + c_{k} x_{k}^{2} + d_{k} x_{k}^{3}, k = 1, 2

(10)

where the unknown eight coefficients $a_{k}, b_{k}, c_{k}, and d_{k}$ of the kth segment can be found by the end conditions together with the continuity conditions given in equation (11)

\begin{matrix} u^{1} (x^{c}) = u_{i}^{2} (x^{c}) \\ u^{1 ″} (x^{c}) = u^{2 ″} (x^{c}) \\ u^{1 ″'} (x^{c}) = u^{2 ″'} (x^{c}) \\ (θ^{2} - θ^{1}) K - EI u^{2 ″'} (x^{c}) = u^{2 ″'} (x^{c}) \end{matrix}

(11)

where $θ$ and $x^{c}$ represent the rotation and the coordinate of crack location, respectively. In this study, crack depth ratio $ξ$ is used to represent the crack severity and taken values between 0 and 1, which represents intact and totally cracked conditions, respectively.

The element matrices were derived following the classical FE procedures based on the assumptions of large transverse displacements, small strains, and moderate to large rotations. Hence, von-Karman kinematic assumptions have been adopted. This model is suitable for beams that elastically deflect an order of its thickness. For the details of the derivation of element matrices, the study by Reddy (2004) can be referred.

Damage detection approach

In this section, two objective functions that rely on optimization techniques have been introduced. The first objective function to be minimized is given as bound constraint nonlinear least square function and based on the periodic solutions of conservative systems

\begin{matrix} f_{1} (θ) = \sqrt{r {(θ)}^{T} r (θ)} : l \leq θ \leq u \\ r (θ) = z_{p 0}^{m} - z_{p}^{c} (z_{p 0}^{m}, T) \\ θ = [L^{c}, α] \end{matrix}

(12)

where $r (θ)$ is the residual function, the vector $z_{p 0}^{m}$ is the maximum amplitudes of measured NNM, $z_{p}^{c} (z_{p 0}^{m}, T)$ is the system response at time T in which it starts motion with initial conditions $z_{p 0}^{m}$ . The parameter vector $θ$ consists of the optimization parameters $L^{c}$ and $α$ , which will be named as crack location and crack severity, respectively. This parameter space is bounded by the upper and lower values $l$ and u. This formulation requires time integration over the period T.

The second objective function is derived based on the stationarity property of potential function $Π$ , given as the sum of strain energy $U_{i}$ stored in system and potential energy function $V_{I}$ of inertia force for a conservative system. As it is well-known, equation (13) takes the form given in equation (14) if a linear vibrating conservative beam is considered. Finally, equation (14) leads to the eigenvalue problem after the application of standard variational operations and FE discretization procedures. Hence, it may be said that the total energy $Π$ given in equation (13) is minimum if the beam vibrates in one of its resonant modes starting its motion with initial conditions that match with the corresponding modal shape

Π = U_{i} + V_{I}

(13)

Π = \frac{1}{2} \int_{0}^{L} EI {(u ″)}^{2} dx - \frac{1}{2} ω^{2} \int_{0}^{L} ρ u^{2} dx

(14)

where the terms $U_{i}$ , $V_{I}$ , EI, $ω$ , $ρ$ , and u in equations (13) and (14) are the strain energy, potential energy function corresponding to inertia force, bending rigidity, circular frequency, mass density, and the displacement function of the beam, respectively. This principle is also true for beams which vibrate nonlinearly at any resonant frequency with initial conditions that match with the corresponding NNM. In this case, energy expression involves the terms from nonlinear restoring force vector $f_{NL}$ together with stiffness matrix K and mass matrix M. In discrete form it can be given as

Π = \frac{1}{2} u^{T} K u + \int_{0}^{u} f_{NL} du - \frac{1}{2} {\overset{\cdot}{u}}^{T} M \overset{\cdot}{u}

(15)

where $u$ is the displacement vector. This approach leads to another objective function that can be effectively used to identify crack locations and magnitudes. If equation (15) is written for zero phase conditions, for example, when initial velocities are zero for each DOF that leads to zero kinetic energy, the second objective function based on minimum strain energy concepts is given by equation (16). Here, superscript N denotes Nth normal mode. This second objective function is computationally inexpensive relative to the first one since it does not require time integration. In addition, the function space contains least local minima as shown in the preceding section

f_{2} (θ) = \frac{1}{2} u_{m}^{N} T K (θ) u_{m}^{N} + \int_{0}^{u_{m}^{N}} f_{NL} (θ, u) du

(16)

The main drawback in both formulations is that the NNM vectors also contain the rotational DOFs, which are difficult to extract from experimental data. Hence, it is mandatory to estimate rotations from displacement components. This can be achieved by following a similar procedure given in Guan and Karbhari (2008).

The displacement function of Nth NNM of each segment of the ith cracked element can be approximated by fourth order piecewise polynomial functions $u_{i}^{1}$ and $u_{i}^{2}$ as given in equation (10). The coefficients of the polynomial functions for each of n number of elements $i = 1, 2, \dots, n$ are given in De Boor (2001) as

\begin{matrix} c_{1, i}^{1} = g (x_{i}) \\ c_{2, i}^{1} = s_{i} \\ c_{3, i}^{1} = ([x_{i}, x_{i}^{c}] g - s_{i}) / (Δ x_{i}) - c_{4, i}^{1} Δ x_{i} \\ c_{4, i}^{1} = (s_{i} + s_{i}^{c} - 2 [x_{i}, x_{i}^{c}] g) / (Δ {x_{i}}^{2}) \\ c_{1, i}^{2} = g (x_{i}^{c}) \\ c_{2, i}^{2} = s_{i}^{2} \\ c_{3, i}^{2} = ([x_{i}^{c}, x_{i + 1}] g - s_{i}^{2}) / (Δ x_{i}^{c}) - c_{4, i}^{2} Δ x_{i}^{c} \\ c_{4, i}^{2} = (s_{i}^{2} + s_{i + 1} - 2 [x_{i}^{c}, x_{i + 1}] g) / ({Δ x}_{i}^{c}^{2}) \end{matrix}

(17)

where $s_{i}$ , $s_{i}^{2}$ , $x_{i}^{c}$ are the rotation at ith node, rotation of the second segment of the element at the crack location and crack location of the ith element, respectively. The divided difference $[x_{i}, x_{i + 1}] g$ is given by

[x_{i}, x_{i + 1}] g = \frac{g (x_{i}) - g (x_{i + 1})}{x_{i} - x_{i + 1}}

(18)

In the presented cubic spline interpolation, there are $4 n - 1$ unknowns, which are rotations $s_{i}^{1}, s_{i}^{2}$ that belong to two segments of the cracked element and the translation $g (x_{i}^{c})$ at crack location together with the rotations $s_{i}$ at the connection of two adjacent cracked elements. These unknowns can be determined from n − 1 equations obtained from conditions given in equations (11) and (19). The unknown rotations calculated in this way are numerically exact since same interpolation functions used in FE formulation have been adopted. Nevertheless, the rotations obtained in this way would differ in the case of noisy NNM measurements which are inevitable in real-life applications

{u^{″}}_{i - 1}^{2} (x_{i}) = {u^{″}}_{i}^{1} (x_{i}) for i = 2 : n - 1

(19)

Any conventional optimization routines can be used to minimize objective functions $f_{1} (θ)$ and $f_{2} (θ)$ . Of course, the unmeasured rotations have to be numerically approximated for different damage states corresponding to $θ$ during iterations. The main issue is that the objective functions $f_{1} (θ)$ and $f_{2} (θ)$ contain several local minima along the crack location $L^{c}$ . However, they contain only a few minima in the parameter space of crack severity $α$ as will be demonstrated in preceding sections. In addition, the function space of $f_{2} (θ)$ is much smoother than the function space of $f_{1} (θ)$ . Unfortunately, $f_{2} (θ)$ is not sensitive to crack severity while it is remarkably sensitive to crack location. Besides, $f_{1} (θ)$ is sensitive to both location and severity. As a result, these properties of the objective functions provide a way to develop an efficient, yet simple identification approach that is summarized by the flowchart given in Figure 1. Briefly, $f_{2} (θ)$ is first minimized to determine crack location by genetic algorithms (GAs) since it is smoother and computationally inexpensive. Subsequently, $f_{1} (θ)$ is minimized to obtain crack severity using a local optimization technique starting from the solution of the previous step.

Figure 1.

Flowchart of the proposed strategy.

Although it is not the case in real-life applications, this method provides the exact crack location and severity if there are no measurement and modeling uncertainties. In order to verify the algorithm in the case of uncertainties, the algorithm was also tested by adding 3% noise to the simulated data using equation (20)

a^{n} = a^{0} (1 + ε C)

(20)

where $a^{n}$ and $a^{0}$ represent the noisy and noise free data vectors, respectively. The variable $ε$ is the noise level and $C$ is a standard random vector subjected to the uniformly distributed random numbers within the interval [−1, 1]. The noise level 3% was adopted based on similar studies in the damage detection literature (Huang et al., 2018; Stutz et al., 2018; Zhang et al., 2018). In this case, both modeling and measurement errors have been considered.

It is noted that no special optimization method was developed for this purpose, instead, the functions in MATLAB were used. The function lsqnonlin, which is used to solve nonlinear least-squares problems, has been used to find the local minimizer of $f_{1} (θ)$ . This method requires the number of optimization parameters to be at least equal to the number of equations (e.g. the problem must not be underdetermined). The function ga was used as the global optimization technique to minimize $f_{2} (θ)$ . This function finds the global minimizer of function, which includes many local minima using GAs. Finally, MATLAB codes were developed by the authors for the cracked beam FE and the arc length continuation method explained in sections “Numerical calculation of NNMs” and “Cracked beam finite element model.”

NNMs of intact and cracked beams

This section provides the frequency–energy plots (FEPs) for the intact and the cracked beam for various crack scenarios. A slender beam with fixed boundary conditions was adopted in the numerical studies. A rectangular cross section of 0.015 m × 0.015 m was chosen. The beam is made of steel with Young’s modulus 210 GPa and mass density of 7890 kg/m³. In a single crack case, the crack location is $L_{1}^{c} = 0.075 m$ . In double crack case, the location of first and second cracks are $L_{1}^{c} = 0.075 m$ and $L_{2}^{c} = 0.675 m$ , respectively. The crack severity $α$ was taken as 0.30, 0.705, and 0.93 in slight, moderate, and severe crack cases, respectively.

A convergence study has been carried out to determine adequate FE mesh. FEP of the first NNM was calculated for various FE meshes. The results from the convergence test given in Figure 2(a) show that the linear frequencies (e.g. frequencies at low energy level or small displacement) do not differ significantly for different number FE elements. However, differences in frequencies increase at higher energy levels. In addition, the secondary branch occurs at different frequencies for each FE mesh. Obviously, the FEPs obtained from the 8 and 10 FE meshes are quite close to each other. Hence, eight FE mesh given in Figure 2(b) was adopted throughout the numerical analysis.

Figure 2.

FE modeling: (a) results from mesh convergence analysis and (b) cracked beam with fixed boundaries.

It should be noted that it is not intended to present full FEPs, for example, the primary and the secondary interaction modes. This is due to the fact that the interaction modes especially the higher ones that emanate from the main branch may be questionable since they may depend on the FE discretization and be affected by the errors during numerical time integration. In addition, their calculation can be very time-consuming and cumbersome due to very small time step requirement. The branch switching phenomena also make it difficult to present all interaction tongues. Nevertheless, the following crack identification study only needs the LNMs and NNMs of the backbone curve (e.g. main branch).

As discussed in section “Numerical calculation of NNMs,” in the calculation of NNMs, the beam vibrates in one of its resonant modes starting its motion with initial conditions $z p_{0}$ that match with corresponding modal shape. Since the system is conservative, for example, there is no energy dissipation, total energy remains constant at the time of the nonlinear time history analysis. Hence, the energy of the beam can be calculated in its initial condition using equation (16), which is actually strain energy of the system. The FEPs of the first and second NNMs of the intact beam are shown in Figure 3. Apparently, the backbones of two NNMs show an increase in frequency as the energy increases. This clearly indicates a stiffening effect on these modes. Besides, an integer ratio of 5:1 internal resonance tongue that shows a modal interaction between NNM1 and NNM2 near 61 Hz has been found. A higher interaction mode has also been identified for NNM2 above 168 Hz.

Figure 3.

FEPs of the intact and slightly cracked beam: (a) first NNM and (b) second NNM.

The FEPs of the first and second NNMs of the slight single and double crack cases are presented in the same plot (Figure 3). It can be observed that the difference in the first and second linear frequencies of intact and slightly cracked beam is not remarkable. This is also true for first frequencies at higher energy levels. Specifically, this difference becomes almost indistinguishable above 5:1 branch. This branch is also observed at almost same energy levels in all cases. Besides, crack introduces a considerable amount of discrepancy in the second modes of the intact and cracked beams. However, the discrepancy in the FEP of single and double cracked beams is too small above frequency of 155 Hz. In other words, second crack does not introduce considerable amount of change in the total energy of the beam.

The FEPs of the first and second NNMs of the cracked beams in the case of the single crack are given in Figure 4. The FEPs were calculated when the crack size is moderate and severe. The first LNM frequencies of the moderately and severely cracked beams are reduced 11% and 21%, respectively. For the second modes, this decrease is around 4.9% and 8.2% for moderate and severe crack cases, which are relatively lower than the decrease in first modes. Besides, different modal interaction tongues are apparent in each FEP. This fact may be an indicator of how the modal interactions are sensitive to damage. At this point, it should be reminded that the modal interactions are also affected by many other factors such as FE discretization and time step size. In addition, their experimental calculation might be so challenging, which makes them not suitable for damage detection studies at least with the current limited knowledge about NNM motion. At first glance, it is not clear in the FEPs of intact and cracked beam if there is a considerable difference as to how fast the frequencies of cracked and intact beams change as the energy change (e.g. slope of backbone). However, a closer look reveals that the average rate of change in frequencies with respect to energy is considerable. The average changes in energy for unit frequency increase are 1.16, 0.76, and 0.51 J/Hz up to the 1.1 times linear frequencies for intact, moderate, and severe cracked cases, respectively. Similarly, average energy changes for unit frequency increase are 2.34, 1.86, and 0.55 J/Hz at higher frequency levels. It is also obvious that the average changes increase due to stiffening affect. This is also true for the second NNMs. Obviously, less energy is needed to create a change in the nonlinear resonance state of the beam when the crack takes place. Although this result is expected, it may be used as an indicator of damage or crack in beams vibrating in the nonlinear regime.

Figure 4.

FEPs in single crack case: (a) first NNM in moderate crack case, (b) second NNM in moderate crack case, (c) first NNM in severe crack case, and (d) second NNM in severe crack case.

Similar observations can be made from the FEPs of the doubly cracked beam given in Figure 5. The decreases in the first LNM frequencies of the beam in moderate and severe double crack cases are 19.67% and 48.65%, while decreases are 8.65% and 20.84% for the second linear mode frequencies, respectively. Various modal interactions tongues that branch off the backbone are encountered in each case. The average change in energy per unit frequency is 0.4951 and 0.3823 J/Hz up to 1.1 times the linear mode frequency, while they are 1.28 and 0.67 J/Hz at frequencies above the 1.2 times the linear mode frequencies.

Figure 5.

FEPs in double crack case: (a) first NNM in moderate crack case, (b) second NNM in moderate crack case, (c) first NNM in severe crack case, and (d) second NNM in severe crack case.

The dependencies of frequencies of the intact and cracked beams on displacement are given in Figure 6 for the first and second NNMs. The stiffening effect is obvious in all cases. Interestingly, curves tend to get close to each other as the amplitude of displacement increases. In other words, the effect of crack on the frequencies decreases. Specifically, stiffening effect is more prominent in severe crack case. The effects of moderate and severe cracks are substantial while it is minor in the slight crack case for the first and the second frequencies. However, the FEPs show that the slight crack generates considerable difference in terms of total energy in the second normal mode (Figure 3). One can realize that this energy difference might arise from the large rotations around crack location; it does not arise from the transverse displacements. Nonetheless, the absolute differences (not the relative ones) in second frequencies of intact and cracked beams are more evident, which makes the higher frequencies more sensitive to damage in every displacement level. It should also be stated that higher frequencies and normal modes usually involve higher uncertainty. The same conclusions can be reached from the NNM motion of the mid-span transverse deflection presented in Figure 6(e) and (f). These NNM motions are given for the same mid-span deflection (0.0082 m) in each case. In slight crack case, the difference in NNM motion of mid deflection is quite low, while it increases as the crack severity increases.

Figure 6.

Frequency–displacement plots: (a) first NNM—single crack case, (b) first NNM—double crack case, (c) second NNM—single crack case, (d) second NNM—double crack case, (e) NNM motion at mid-span (translation) in single crack case, and (f) NNM motion at mid-span (translation) in double crack case.

Numerical crack detection study

Identification without modeling and measurement uncertainty

The effectiveness of the approach in detecting both crack location and severity was investigated using both LNMs and NNMs. At first, the objective function space was investigated in the case of single damage. Subsequently, identification results were presented. In all cases, only first LNMs and NNMs were used. It was assumed that the model and the data are noise free.

The function space of the first objective function $f_{1} (θ)$ in the case when there is one moderate single crack is visualized in Figure 7. The solution space for LNM at 49.24 Hz and NNM at 57.42 Hz is quite similar as given in Figure 7(b) and (c). Apparently, the objective function reaches its minimum where the crack is located. In addition, minimum is achieved when the crack severity takes its exact value, which is $α = 0.705$ in this case. Hence, it can be said that the first objective function can be used to exactly estimate both the crack location and the severity when the data are noise free. That is to say, $f_{1} (θ)$ is sensitive to both crack location and severity. However, it is obvious that $f_{1} (θ)$ contains many local minima along the crack location coordinate, which makes it very difficult to detect the optimal values by local optimization algorithms. In addition, this function also contains local minima at the element level along the crack location. Nevertheless, objective function is much more smooth, for example, contains only two minimum along crack severity parameter.

Figure 7.

Plot of $f_{1} (θ)$ with respect to crack severity and location in the case of moderate single damage: (a) solution space in the case of LNM is used, (b) solution space in the case of NNM is used, (c) objective function value—crack severity at L = 0.675, and (d) objective function value—crack location for $α$ = 0.705.

The plot of $f_{1} (θ)$ in the case of single slight crack given in Figure 8(a) shows that the solution space is smoother than the previous one and the number of local minima is substantially less. This would provide the opportunity to detect slight cracks more precisely if noise does not exist. In other words, the probability to find the global minimizer of $f_{1} (θ)$ would increase. In addition, the objective function contains only one minimum along the crack severity axis for crack severity smaller than 0.5, which can be found by using local optimization.

Figure 8.

Plot of objective function value with respect to crack severity and location in the case of single slight crack: (a) solution space of $f_{1} (θ)$ , (b) $f_{1} (θ)$ value—crack severity for L = 0.075, (c) $f_{1} (θ)$ value—crack location for $α$ = 0.3, (d) solution space of $f_{2} (θ)$ , (e) $f_{2} (θ)$ value—crack severity for L = 0.075, and (f) $f_{2} (θ)$ value—crack location for $α$ = 0.705.

The second objective function $f_{2} (θ)$ given in Figure 8(d) that is based on the strain energy minimization is much smoother than $f_{2} (θ)$ . In addition, it contains only one minimum at element level along the beam length. Although the minimization of $f_{2} (θ)$ also requires global minimization methods in order to find the crack location, the convergence would be faster. This function reaches its minimum at the crack location close to the exact one. However, the disadvantage of $f_{2} (θ)$ is that this function is not sensitive to the crack severity. In other words, it does not take a minimum value for the exact value of $α$ = 0.93 as shown in Figure 8(e). Instead, the minimum is achieved when $α$ = 0 if LNM is used (e.g. f = 43.43 Hz) and $α$ = 0.2 if NNM at 51.35 Hz frequency level is used.

In view of the previous discussion, one can benefit from two objective functions for efficient crack identification. The use of $f_{2} (θ)$ in conjunction with $f_{1} (θ)$ provides an efficient way to obtain the correct crack parameters as discussed in section “Damage detection approach.” In order to verify the accuracy and the efficiency of the presented approach, numerical results are provided for the aforementioned crack scenarios. Since the GA is a class of metaheuristic optimization and a stochastic one, it has run several times to verify its robustness. In the first step, GA has run 10 times and the identified parameters are given in Figure 9 in the case of single severe crack for LNM and NNM at 51.35 Hz, respectively. In this case, population size was taken as 20 and it was taken as 60 in the double damage case in which there are two and four optimization parameters, respectively. Results clearly show that the crack location is identified almost exactly in all runs while different values for the crack severity were obtained. This is due to the fact that this objective function is very sensitive to crack location as discussed previously. In all runs, the crack was immediately identified after 20 function evaluations. The results do not differ significantly in the case when LNM or NNM is used. The crack location was identified falsely only in a run when LNM was used. However, the error in identified location is quite low. In this run, the objective function value is considerably higher as shown in Figure 9(c). This could be due to the premature convergence of the GA population. In these cases, the population size can be increased freely since $f_{2} (θ)$ is computationally very inexpensive.

Figure 9.

GA results from the minimization of $f_{2} (θ)$ in single severe crack case: (a) identified crack locations, (b) identified crack severity, (c) minimum objective function values for each run when LNM is used, and (d) minimum objective function values for each run when NNM is used.

The identified crack parameters from the minimization of $f_{2} (θ)$ for other crack scenarios are presented in Table 1. As seen, the identified crack locations are fairly accurate. Maximum error in crack location is 24% among all cases and belongs to the single slight crack case when NNM at 62.27 Hz is used. This error can arise due to the fact that the sensitivity of $f_{2} (θ)$ to crack location decreases when higher NNMs are used. Specifically, first NNMs at higher frequencies are less sensitive to cracks as discussed in section ‘NNMs of intact and cracked beams’. Besides, second NNMs show a remarkable change in terms of total energy due to the existence of cracks. However, this change in transverse displacements is low. The crack locations identified in double crack cases are more accurate due to the fact that the LNMs and NNMs of doubly crack beams are more distinctive properties, for example, they are more sensitive to crack location. Similar to single crack case, the sensitivity of $f_{2} (θ)$ to slight crack is lesser than moderate and severe crack cases. Nevertheless, the identified crack severities are incorrect in all cases as expected. The minimum objective function values, which correspond to minimum strain energy, are given in the last column of Table 1. These values demonstrate that strain energy decreases with the increase in crack number and severity. Hence, the possibility of the detection of cracks increases.

Table 1.

Identified crack parameters using $f_{2} (θ)$ (the best run out of 10 runs).

Damage scenario	Normal mode motion	First crack location $L_{1}^{c}$ (m)	Second crack location $L_{2}^{c}$ (m)	Crack severity $α_{1}$	Crack severity $α_{2}$	Best objective function value
Single slight crack	LNM (54.70 Hz)	0.0650	–	0.86	–	5.74e–5
	NNM (62.27 Hz)	0.0567	–	0.92	–	8.74
Single moderate crack	LNM (49.24 Hz)	0.0693	–	0.1801	–	4.41e–5
	NNM (57.42 Hz)	0.0603	–	0.9258	–	7.8603
Single severe crack	LNM (43.43 Hz)	0.0717	–	0.1059	–	3.289e–4
	NNM (51.35 Hz)	0.0621	–	0.9094	–	4.8838
Double slight crack	LNM (54.27 Hz)	0.0720	0.651	0.903	0.96	4.8e–5
	NNM (62.61 Hz)	0.0627	0.644	0.85	0.71	8.70
Double moderate crack	LNM (44.45 Hz)	0.0763	0.6760	0.1289	0.1891	2.86e–5
	NNM (54.75 Hz)	0.0671	0.6699	0.8311	0.6072	7.796
Double severe crack	LNM (28.41 Hz)	0.0765	0.6747	0.9320	0.1981	1.20e–4
	NNM (39.57 Hz)	0.0748	0.6725	0.8579	0.2563	2.6018

LNM: linear normal mode; NNM: nonlinear normal mode.

In all cases, the exact crack locations and severities have been found in the second step where $f_{1} (θ)$ has been used since the data are noise free. However, the number of function evaluations is different for all cases. Thus, the evaluation of objective function is given in Figure 10. In severe crack case, maximum 40 function evaluations are needed to obtain true crack parameters. In addition, local optimization was used starting from the initial conditions obtained from the first step. Since the crack severities obtained from the first step are incorrect, it is suitable to choose $α = 0$ as initial crack severity. In slight damage case, the required number of function evaluations is high especially in the case where NNMs are used. In this case, the local optimization has not succeeded and GA was used instead. This was also observed in moderate crack case, and substantial number of function evaluation was required to achieve the global minimum even though the initial crack locations identified from $f_{1} (θ)$ were quite close to true ones. Consequently, one can say that the adopted strategy successfully identified the crack parameters in all cases by using only first LNM and/or NNM. It may also be said that the capability of the method can be increased by adding higher order normal modes as demonstrated by previous studies in FE model updating literature. However, measuring higher order modes is difficult for real-life structures.

Figure 10.

Convergence graphs: (a) single slight crack, (b) double slight crack, (c) single moderate crack, (d) double moderate crack, (e) single severe crack, and (f) double moderate crack.

Identification with modeling and measurement uncertainty

In order to verify the capability of the approach in crack identification, modeling and measurement errors, which inherently exist in real-life applications, are taken into account. Initially, modeling errors have been taken into account and the same calculations have been carried out. In this case, the FE model used in calculations is composed of four elements. Thereafter, the data have been contaminated by adding 3% random noise to the data according to equation (20).

The results obtained from the analysis in the case of only modeling errors are presented in Table 2. The results are fairly good in all single crack cases. The estimation error in crack location is below 0.4% for $f_{1} (θ)$ , while it is slightly higher for $f_{2} (θ)$ . This is also true for the estimations of crack severities. The estimates of crack severities for $f_{2} (θ)$ were not given since they were falsely identified in all cases as discussed previously. The percentage error in crack severities is less than 5% in all single crack cases. Maximum error was observed in slight crack case. The number of function evaluations is quite low for slight and moderate cases since local optimization succeeded in finding the correct minimizer. However, the function evaluation is around 320 for severe case since GA was used in this case since the initial crack location $(L^{c} = 0.0165)$ obtained from the first step is far from the exact one $(L^{c} = 0.075)$ .

Table 2.

Crack identification results in the case of modeling errors ( $(L_{true}^{c} - L_{predicted}^{c}) / L$ and percentage errors in $α$ are given in parenthesis).

Damage scenario	First crack location, $L_{1}^{c}$ (m)		Second crack location, $L_{2}^{c}$ (m)		Crack severity, $α_{1}$	Crack severity, $α_{2}$	Number of function evaluations
Damage scenario	$f_{2} (θ)$	$f_{1} (θ)$	$f_{2} (θ)$	$f_{1} (θ)$	$f_{1} (θ)$	$f_{1} (θ)$	$f_{1} (θ)$
Single slight crack	0.081 (0.54%)	0.079 (0.39%)	–	–	0.3158 (5.3%)	–	33
Single moderate crack	0.049 (2.12%)	0.075 (0.00%)	–	–	0.7091 (0.58%)	–	21
Single severe Crack	0.016 (4.88%)	0.075 (0.00%)	–	–	0.933 (0.32%)	–	320
Double slight crack	0.089 (1.24%)	0.498 (35.28%)	1.114 (36.2%)	0.668 (0.56%)	0.38 (26.67%)	0.36 (20%)	660
Double moderate crack	0.003 (6.00%)	0.091 (1.38%)	0.604 (5.89%)	0.678 (0.32%)	0.7364 (4.45%)	0.7789 (10.4%)	360
Double severe crack	0.057 (1.5%)	0.050 (2.06%)	0.658 (1.42%)	0.660 (1.25%)	0.826 (11.18%)	0.926 (0.43%)	30

In double crack cases, the errors are higher than the errors observed in single crack case, yet, still acceptable. Error in crack locations is less than 6% and 2% for $f_{1} (θ)$ and $f_{2} (θ)$ , respectively, in moderate and severe crack cases. As expected, the errors are considerably high in slight damage case due to the decrease in the sensitivity of the objective function. The function evaluation is higher in double crack case. GA was used in all cases except the double severe case. Besides, identified crack severities are substantially good in all cases.

The identified parameters in the case of modeling and measurement errors are presented in Figure 11 for double crack case only. The results are also similar for single damage case. The method still generates good results despite the uncertainties in both the model and data. In addition, two objective functions provide satisfying results. On the contrary, one of the crack locations was falsely identified due to noise in slight crack case as in the previous case. However, the identified crack severity is fairly good. Apparently, it is clear that the proposed method is not remarkably affected by the uncertainties, specifically when the crack is severe. Accuracy decreases as the crack severity decreases.

Figure 11.

Identification results in double crack case when there are 3% noise and modeling errors: (a) first crack location, (b) second crack location, and (c) crack severities.

Conclusion and discussion

A crack detection approach which can be applicable to beams vibrating in either linear or nonlinear regime was introduced. The approach was developed using LNMs and NNMs of conservative systems and formulated as a minimization problem based on two objective functions.

The presented results from the several numerical crack cases clearly showed that the damage identification strategy is well capable of detecting both crack location and severity. It was realized that the second objective function, which is based on minimum strain energy, is sensitive to crack location, while it is computationally very inexpensive. On the contrary, the first objective function that is based on the periodic solutions of conservative mechanical systems is sensitive to both crack location and severity. However, this function is computationally expensive relative the second one since it requires time integration. Besides, the sensitivity of these objective functions decreases when the damage is slight.

Consequently, the use of normal mode approach in challenging crack detection problems was shown to be encouraging. It was also shown that the NNMs of the geometrically nonlinear beams are promising to be used in damage/crack identification problems in the case when nonlinear vibration data are available. Improvement in measurement systems and computational techniques would increase the potential of the presented approach in complex real-life problems. Although the results obtained using numerically simulated data are promising, the algorithms should be tested using experimental data from carefully designed test campaigns as a future work.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yildirim Serhat Erdogan

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